Enhanced performance of mixed HWMA-CUSUM charts using auxiliary information

Quality control (QC) is a systematic approach to ensuring that products and services meet customer requirements. It is an essential part of manufacturing and industry, as it helps to improve product quality, customer satisfaction, and profitability. Quality practitioners generally apply control charts to monitor the industrial process, among many other statistical process control tools, and to detect changes. New developments in control charting schemes for high-quality monitoring are the need of the hour. In this paper, we have enhanced the performance of the mixed homogeneously weighted moving average (HWMA)-cumulative sum (CUSUM) control chart by using the auxiliary information-based (AIB) regression estimator and named it MHCAIB. The proposed MHCAIB chart provided an unbiased and more efficient estimator of the process location. The various measures of the run length are used to judge the performance of the proposed MHCAIB and to compare it with existing AIB charts like CUSUMAIB, EWMAAIB, MECAIB (mixed AIB EWMA-CUSUM), and HWMAAIB. The Run length (RL) based performance comparisons indicate that the MHCAIB chart performs relatively better in monitoring small to moderate shifts over its competitor’s charts. It is shown that the chart’s performance improves with the increase in correlation between the study variable and the auxiliary variable. An illustrative application of the proposed MHCAIB chart is also provided to show its implementation in practical situations.


Introduction
Statistical process control and monitoring (SPCM) consists of several statistical tools, and control charts are considered the most efficient.The control charts resolve irregular deviations from the required standards in manufacturing and industrial processes.The memory-less and memory types are the two core divisions of the control charts (cf.Montgomery [1]).Shewhart [2] proposed memory-less control charts, which use only current sample information for process monitoring.The memory type charting procedures, for instance, the cumulative sum (CUSUM), the exponentially weighted moving average (EWMA), the progressive mean (PM), and the homogeneously weighted moving average (HWMA) were developed by Page [3], Roberts [4], Abbas et al. [5] and Abbas [6] respectively and the monitoring statistics of these charts grasp earlier sample information along with the recent information.
On control charts, various types of extensions have been introduced in the literature on the SPCM.Combining two control charts also improved the efficiency of the control charts.Lucas [7] and Lucas and Saccucci [8] suggested the combined design structure of the Shewhart-EWMA and Shewhart-CUSUM charts, respectively.Shamma and Shamma [9] proposed a double EWMA chart.Mixed design structures of EWMA-CUSUM (MEC) and CUSUM-EWMA (MCE) charts were suggested by Abbas et al. [10] and Zaman et al. [11] respectively.Motivated by the study of Shamma and Shamma [8], double PM and HWMA charts were suggested by Abbas et al. [12] and Abid et al. [13] respectively.A mixture of PM and EWMA charts was proposed by Abbas et al. [14].Taking inspiration from Abbas et al. [9] and Abid et al. [15] developed a mixed HWMA-CUSUM (MHC) chart in which statistic of the CUSUM chart runs as the output and MHC chart outperforms against the EWMA, HWMA, and MEC charts.
In sample surveys, the precision of the estimates of the population parameters can be increased by using auxiliary information.The auxiliary variable is a variable known for all units of the population but not a variable under study.The auxiliary information-based (AIB) charts are usually developed using regression and ratio estimators to monitor the process variable effectively.In the SPCM literature, much work has been done related to the AIB charts.Riaz [16]and Riaz [17] proposed a regression estimator-based Shewhart (AIB) chart for monitoring process location and dispersion, respectively.The regression EWMA AIB chart was proposed by Abbas et al. [18] and the EWMA AIB performed well against the usual EWMA chart without the AIB information.Abbas [19] suggested the CUSUMAIB chart performed relatively better than the usual CUSUM chart.Ahmad et al. [20] suggested some AIB charts for the autocorrelated processes.Adegoke et al. [21] designed a regression HWMA AIB chart when the process variable is investigated under normal and non-normal environments and revealed that the HWMA AIB chart is more powerful than the EWMA AIB and CUSUM AIB charts.Sanusi et al. [22] suggested various ratio estimators based on EWMA charts.The regression PM AIB chart was suggested by Abbas et al. [12] under zero-state and steady-state processes.Interested readers can see the work of Ahmad et al. [20], Haq and Khoo [23], Abbasi and Haq [24], Noor-ul-Amin et al. [25], and Hussain et al. [26] on AIB charts.Dirbaz et al. [27] suggested two new AIB-based control charts, AIB-MEWMA and AIB-DMEWMA charts, to detect shifts in model parameters.Arslan et al. [28] designed a sensitive homogeneously weighted moving average chart using two supplementary variables (hereafter, TAHWMA), which is an efficient and unbiased estimator for the process mean if the two supplementary variables correlate with the study variable.
In the SPCM literature, very little work is available on AIB mixed memory control charts.Recently, Anwar et al. [29] designed a regression estimator based MEC AIB and MCE AIB charts for prompt detection of persistent changes, and the MEC AIB chart is more effective than the MCE AIB chart and as well as the EWMA AIB and CUSUM AIB charts.Adegoke et al. [21] revealed that the performance of the regression estimator is relatively better than the ratio estimator.The core focus of this study is to propose an efficient mixed memory chart under the scenario of the regression estimator.So, this study proposes a new regression estimator based MHC chart labeled as MHC AIB for detecting persistent deviations in the process location.The MHC AIB chart is a mixture of the HWMA AIB and the usual CUSUM chart.In the recent age of development, improvements in quality assurance techniques are the need of the hour.In context, we have developed a new chart showing visible improvement in detecting shifts.Even a minute change and deviation in the quality can be a big hurdle in many industrial processes like lifesaving drugs, substrate manufacturing, missile equipment, etc.In some industrial and manufacturing processes, the auxiliary information is also recorded along with the understudy variables for different tasks.This information can be used to improve the control chart design without imparting any additional financial burden to the entrepreneur.We can use this information for the improvement of design.Our proposed chart is shown to have improved results compared to its counterparts and can be used for high-quality monitoring in different industrial applications.
The rest of the article is outlined as follows: the next section offers the structure of the MHC and the MHC AIB charts, along with the RL evaluation of the proposed MHC AIB chart.The performance evaluation and RL comparisons of the MHC AIB chart against the competitor's charts are delivered in Section 3. A numerical example of the MHC AIB and existing charts are offered in Section 4, Section 5 gives the limitation of the study, and the article ends with a conclusion and recommendations.

The Mixed HWMA-CUSUM (MHC) and the proposed Mixed HWMA-CUSUM with auxiliary information (MHC AIB ) control charts
This section includes a description of the construction of the existing Mixed HWMA-CUSUM (MHC) and the proposed Mixed HWMA-CUSUM with additional information MHC AIB charts:

The MHC chart
Let z ij is the variable of interest follows the normal distribution, i.e., z ij e N m z ; s 2 z À � where μ z and s 2 z is the in-control (IC) mean and variance of the process variable, respectively, i = 1, 2, 3, . . .and j = 1, 2, 3, . .., n.Abbas [6] suggested the statistic of the HWMA chart as follows: Where θ is the smoothing parameter (θ 2 (0, 1]), � � z 0 ¼ m z , and � � z iÀ 1 ¼ . The IC mean and variance for H i are as follows (cf.Abbas [6]): Abid et al. [15] proposed the MHC chart by placing the statistic given in (1) with the CUSUM statistic, and the plotting statistics of the MHC chart are given as: Where H i is given in (1) and The K and H are defined as (cf.Abid et al. [15]) ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi And these are the parameters of the MHC chart.The process is considered to be out-ofcontrol (OOC) if any value of MHC þ i or MHC À i go beyond H; otherwise, it is considered to be in control.

The proposed MHC AIB chart
In most situations, there exists a positive/negative association between the study/process variable (z i ) and the auxiliary variable(x i ).Let us assume that x i is strongly correlated with z i and the strength of the correlation between x i and z i is represented by ρ zx .The pair of observations x i and z i follows the bivariate normal distribution, i.e., (z i , x i ) ~BVN(μ z + δσ z , μ x , σ z , σ x , ρ zx ), where δ is mathematically written as d ¼ ffi ffi n p s z jm z À m 1 j, where μ 1 is the shifted mean.The regression estimator suggested by Cochran [30] is as follows: where b zx ¼ r zx s z s x is the regression coefficient, � z i and � x i is the sample mean of the z i and x i , respectively, and μ x is the population mean of the x i .The mean and variance of R i are given below (cf.Appendix A in S1 File): The plotting statistic of the proposed MHC AIB using ( 2) is defined as: Where The MHC AIB chart depends on two parameters, i.e., K and H and which are mathematically written as: (cf.Abid et al. [15]): ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi RHC þ i and RHC À i are plotted against the value of H given in (10).
i > H, the process is assumed to be out of control (OOC); otherwise, it is in control (IC).

Performance evaluation of the MHC AIB chart
The help of a well-known measure assesses the performance of the proposed MHCAIB and its competitor charts called the average run length (ARL).The Average Run length can be defined as "The number of sample points before a control chart gives alarm is called run length (RL) and an average value of RL distribution is called ARL."The ARL IC and ARL OOC designated the in-control (IC) and out-of-control (OOC) ARL.A chart having a lesser value of ARL OOC for a particular shift is designated to be better over the competitor chart at a certain shift in the process parameter(s) for a fixed value of ARL IC .We have also included some other performance measures associated with RL, like (the standard deviation RL (SDRL) and the median RL (MDRL) (cf.Abid et al. [15])) and these measures are calculated through the Monte Carlo simulations approach.The computational algorithms are developed in the R programming language, and the computational algorithms' flow chart is presented in   4)).An increase in ρ zx enhanced the efficiency of the MHC AIB chart (for instance δ = 0.05, θ = 0.1 when ρ zx = 0.25, ARL OOC = 117.43against ρ zx = 0.95, ARL OOC = 39.74 (cf.Table 1)).There is a decrease in the OOC SDRL and MDRL values with the increase in ρ zx (for instance δ = 0.125, θ = 0.1 when ρ zx = 0.25, OOC SDRL = 71.08,OOC MDRL = 22 against ρ zx = 0.95, OOC SDRL = 11.66,OOC MDRL = 12 (cf.Table 1)).The performance assessment of the MHC AIB chart in the form of a line graph is given in

Comparisons
This section offered the OOC performance assessment of the proposed MHC AIB with the CUSUM AIB , EWMA AIB , HWMA AIB , and MEC AIB suggested by Abbas et al. [18], Abbas [19], Sanusi et al. [22], and Anwar et al. [27] respectively.Moreover, we have also expressed these comparisons as a percentage decrease in ARL.(ARL PD ) and mathematically ARL PD is defined The chart with the highest ARL PD value is labeled an efficient chart for that specific shift.

Graphical comparisons based on ARL OOC
The ARL OOC based graphical comparisons of the proposed MHC AIB , EWMA AIB , HWMA AIB , and MEC AIB charts are presented in

An illustrative example
Based on the simulated dataset, this section offers an illustrative example of the proposed MHCAIB, HWMAAIB, and MECAIB charts.This dataset consists of 20 pairs of observations which are obtained from the bivariate normal distribution, i.e., (z i , x i ) ~N(μ z + δσ z , μ x , σ z , σ x , ρ zx ) by using the following values of μ z = 0, δ = 0.50, μ x = 0, σ z = 1, σ x = 1 and ρ zx = 0.50 (cf.Abbas et al. [18]).This dataset is used to evaluate the shift-detecting capability of the proposed MHC AIB , HWMA AIB , and MEC AIB charts.The selected parameters for the practical implementation of the proposed MHC AIB , HWMA AIB , and MEC AIB charts are as follows: for the proposed MHC AIB chart θ = 0.1, k = 0.5, and h = 8.575; for the MEC AIB chart θ = 0.1, k = 0.5, and h = 37.35; for HWMA AIB chart θ = 0.1, and C = 2.936 when ARL IC � 500.The control limits and the plotting statistics of the MHC AIB , HWMA AIB , and MEC AIB charts are given in Table 7.
The HWMA AIB chart signals only one OOC point at the 18 th sample (cf.Fig 4).The MEC AIB chart cannot produce any OOC signal (cf.Fig 5).Moreover, the proposed MHC AIB chart produces nine OOC signals from sample numbers 12 to 20 (cf.Fig 6), and this is a piece of evidence about the enhanced shift-detecting capability of the proposed MHC AIB chart against the HWMA AIB , and MEC AIB charts.

Limitation
The proposed chart uses auxiliary information in its design, so it should be used only if there is a high correlation between the auxiliary variable and the study variable.

Conclusion and recommendations
A control chart is the most famous statistical process control and monitoring tool to detect irregular variations from ongoing processes.In this article, we have suggested a new regression estimator-based MHC chart labeled MHC AIB for monitoring persistent deviations in the process location.The ARL OOC performance of the suggested MHC AIB chart is compared with the CUSUM AIB , EWMA AIB , HWMA AIB , and MEC AIB , and the suggested MHC AIB chart performs  exceptionally well in detecting shifts over its competitor charts for all selected sets of θ and ρ zx .Also, it is noticed that the choice of the larger value of ρ zx and a smaller value of θ is effective in enhancing the performance of the MHC AIB chart.An application based on simulated data has also identified the dominance of the MHC AIB chart against the HWMA AIB and MEC AIB charts.This study can also be extended for dual auxiliary information-based regression estimator for detecting deviations in the process location and dispersion.

Fig 1 .
Fig 1.The computational algorithm of the proposed MHC AIB chart.https://doi.org/10.1371/journal.pone.0290727.g001 Fig 2A-2D against various choices of θ and ρ zx .A decrease in θ enhanced the efficacy of the MHC AIB chart and vice versa (cf.Fig 2A vs. Fig 2D).Also, an increase in the value of ρ zx improved the sensitivity of the MHC AIB chart and vice versa.It means that a higher correlation coefficient value increases the suggested chart's efficiency.(cf.Fig 2A-2D).

Table 1
vs. Table

MHC AIB versus MEC AIB .
Anwar et al. [29]suggested the MEC AIB chart and the results of ARL OOC values of MEC AIB chart are specified in Tables